Comptes Rendus
no. 17-18
Complex Analysis
Oka manifolds
Mikhaël Gromov
Franc Forstnerič1
1Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1017-1020

We give a positive answer to Gromov's question [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881]: If every holomorphic map from a compact convex set in a Euclidean space Cn to a certain complex manifold Y is a uniform limit of entire maps Cn→Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion O(X,Y)↪C(X,Y) of the space of holomorphic maps into the space of continuous maps is a weak homotopy equivalence.

Nous donnons une réponse positive à la question suivante posée par Gromov [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881] : Si une variété analytique complexe Y est telle que toute application holomorphe d'un voisinage d'un sous-ensemble compact convexe de l'espace euclidien Cn dans Y peut être approximée par des applications entière de Cn dans Y, alors les applications holomorphes d'un espace de Stein réduit X dans Y possèdent la propriété d'Oka paramétrique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.07.005

Franc Forstnerič  1

1 Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia 
Franc Forstnerič. Oka manifolds. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1017-1020. doi: 10.1016/j.crma.2009.07.005
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